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The simplicial identities encode the relationships between the face and degeneracy maps in a simplicial object, in particular, in a simplicial set.
The simplicial identities are the duals to the simplicial relations of coface and codegeneracy maps described at simplex category:
Let sSet with
face maps obtained by omitting the th vertex;
degeneracy maps obtained by repeating the th vertex.
The simplicial identities satisfied by face and degeneracy maps as above are (whenever these maps are composable as indicated):
(e.g. Goerss & Jardine 1999/2009, I.1. (1.3))
The middle case in (1) implies in particular that degeneracy maps are (split) monomorphisms: One may think of them as including degenerate simplices into all simplices of a given dimension.
The simplicial identities of def. can be understood as a non-abelian or “unstable” generalization of the identity
satisfied by differentials in chain complexes (in homological algebra).
Write be the simplicial abelian group obtained form by forming degreewise the free abelian group on the set of -simplices, as discussed at chains on a simplicial set.
Then using these formal linear combinations we can sum up all the face maps into a single map:
The alternating face map differential in degree of the simplicial set is the linear map
defined on basis elements to be the alternating sum of the simplicial face maps:
This is the differential of the alternating face map complex of :
The simplicial identity def. (1) implies that def. indeed defines a differential in that .
By linearity, it is sufficient to check this on a basis element . There we compute as follows:
Here
the first equality is (2);
the second is (2) together with the linearity of ;
the third is obtained by decomposing the sum into two summands;
the fourth finally uses the simplicial identity def. (1) in the first summand;
the fifth relabels the summation index by ;
the last one observes that the resulting two summands are negatives of each other.
For original references see at simplicial set.
Review includes:
Peter May, Def. 1.1 in: Simplicial objects in algebraic topology, University of Chicago Press 1967 (ISBN:9780226511818, djvu, pdf)
Paul Goerss, J. F. Jardine, Eq. (1.3) in Section I.1 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4, webpage)
Last revised on April 19, 2023 at 19:07:09. See the history of this page for a list of all contributions to it.